A Category of Probability Spaces

TL;DR

This paper develops a categorical framework for probability spaces, introducing a functor that generalizes conditional expectations and explores their properties within this structure.

Contribution

It defines a category of probability spaces with measurable functions, introduces a functor for conditional expectations, and studies their interaction and the completion process.

Findings

Conditional expectations are generalized as functorial constructs.

The interaction between f-measurability, f-independence, and conditional expectations is analyzed.

The completion of probability spaces is formulated as an endofunctor of Prob.

Abstract

We introduce a category Prob of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving direction of information with a way of its interpretation. We introduce a contravariant functor E from Prob to Set, the category of sets. The functor E provides conditional expectations along arrows in Prob, which are generalizations of the classical conditional expectations. For a Prob arrow f, we introduce two concepts f-measurability and f-independence and investigate their interaction with conditional expectations along f. We also show that the completion of probability spaces is naturally formulated as an endofunctor of Prob.